Hidden Harmonies

by Ellen Kaplan

  He then went to work with similar triangles, the ratios of their sides, and the fact that if two angles inscribed in a circle cut off the same arc, they are equal.

  This is how he did it.

  —quod fecit Ptolemaeus?p

  Ptolemy needed this theorem to deduce the key trigonometric formulae for sines and cosines of angle sums and differences—but he brought someone else home as well from his circle dance, since the Pythagorean Theorem follows when the inscribed quadrilateral is a rectangle: for now AB = CD, AD = BC, and AC = BD, so

  SHAPES OTHER THAN SQUARES

  In Chapter Four you saw how Euclid extended the Theorem from squares to any similar polygons on a right triangle’s three sides. More than a century before him, Hippocrates of Chios was thinking about semicircles on those three sides—for these too will be similar (all semicircles are). He used this fact in what looks like an attempt to square the circle.

  It was he who, in Chapter Three, came up with two mean proportionals in hopes of doubling the cube. The Greeks were equally troubled by how to make (with straightedge and compass) a circle that had the same area as a given square, and his idea may have been that forming a controlled part of a circle with the same area as a right triangle would have been a sizable stride along the way. The part he found was a lunule, a quarter moon—

  and while this didn’t turn out to help with squaring the circle (nothing would have: it took twenty-two hundred years to prove that the circle couldn’t be squared), it did leave us this attractive consequence of the Pythagorean Theorem:

  THEOREM: On right triangle ABC construct semicircles on sides AC and BC, and inwardly on AB, giving us the regions labeled Q and R in this diagram. Then | ABC | = | Q | + | R |.

  PROOF:

  Squares, polygons, semicircles—aren’t we about to swing open the shutters again? For as long as a figure has area, that of those similar to it will vary as the square of corresponding linear measures—in our case, the line-segments attached to our triangle’s three sides. What you’re seeing is the continuing abstraction from ‘square’ as denoting a shape, to connoting the measure of area per se. This process was under way several chapters back, when we spoke of a shape not being but having a square.

  Read in this way, the Pythagorean Theorem therefore holds even when the most art nouveau shapes flourish on a right triangle’s hypotenuse, along with shapes similar to it on the legs. They can, if you wish, be as lacy as your great-grandmother’s antimacassars, so long as they have areas:

  The wonders of calculus allow us to determine the areas of such lacework (the integral began its life with just this aim), by extending the notion of triangulating you saw in Chapter Four—dividing and conquering to the limit.q At the same time, it extends the notion of what can have area—but only so far. Too many holes, too close together, and area evaporates. That would be true, for instance, of the famous Sierpin-ski Gasket, made by deleting the ‘midpoints triangle’ from an original triangle, then the midpoints triangles from those remaining—and so on, forever. The area of what’s left turns out to be zero (a quarter of what was

  there at each stage is removed at the next). Yet since 0 + 0 = 0, you could, in a frolicsome mood, extend the Pythagorean Theorem to a right triangle sporting Sierpinski Gaskets on its sides. Or has our drama just turned absurdist?

  TRIANGLES OTHER THAN RIGHT

  Dog owners grow to resemble their dogs, and mathematicians the abstract pets they care for. They even lose their specific names and become generic, as did Pappus. What we have of him is that surname, chunks of his Synagoge (The Collection), and the date of an eclipse he mentions: A.D. 320. He taught in Alexandria, and broadened the Pythagorean Theorem to all triangles by proving that if parallelograms were built on two sides, there would be a parallelogram on the third side whose area was the sum of theirs.

  Here is how an ironically modern turn of thought might show this. The area of those two given parallelograms is some positive number, call it A. On the third side let a parallelogram grow steadily larger, from an area of zero (when, collapsed, its long sides coincide) to a size with area well past A. Somewhere along the way will have been the parallelogram you sought, with area exactly A.

  You should feel as cheated by this as you would were you to ask a stranger whether he could give you directions to a renowned restaurant, and he answered that indeed he could, and walked away. “I meant . . . ,” you shout after him, in vain. That’s the last time you’ll ever visit this city.

  Pappus had no irony in his soul. These are his directions for building exactly the renowned parallelogram you hoped for. Everything turns on transitivity among parallelograms.

  Starting with the same givens, ACB and the parallelograms ADEC and BCGF on two of its sides, extend the upper side, DE, of one parallelogram to meet the extension of FG, the upper side of the other, at H, and extend HC to meet AB at K. Construct AL and BM parallel to HK, meeting DE and FG at L and M respectively, and draw LM, meeting HK at N. ALMB is the desired parallelogram.

  For since ALHC and BMHC are parallelograms, sharing side HC, AL BM and AL || BM, so ALMB is indeed a parallelogram.

  Now | ADEC | = | ALHC |, since they share the base AC and lie between the same parallels, DE and AC.

  But | ALHC | = | ALNK |, since they share the base AL and lie between the same parallels, AL and KH.

  By transitivity, | ADEC | = | ALNK |, and one of our original parallelograms has the same area as part of the desired one.

  In the same way, | BFGC | = | BMNK |: the second of the original parallelograms has the same area as the remainder of the desired one, and we’re done.

  Pappus’s Theorem not only generalizes the Pythagorean but—like Ptolemy’s—generates it, since it is proven independently (and the Pythagorean follows when C is taken as a right angle and the parallelograms as squares). It is its own grandpa. Those nested cones of generalization we spoke of—how long the view down them in either direction, reminding us that in math, as in most things, we always stand midmost. Unlike our simple model of causality, however, these endless cones needn’t lie in a straight line: their array might, as here, cavort in self-intersections.

  We mentioned that the Pythagorean Theorem startles us in part because we estimate area so badly: the square on the hypotenuse still doesn’t look big enough to contain the squares on the two sides. What would you make of someone who saw that in any triangle, the sum of the squares on the two sides added up to twice the square on half of the third, plus twice the square on the median to that side? “Prodigy” comes to mind, with its hint of the uncanny. Apollonius of Perga, then, was such a prodigy, for this, they say, is his theorem. He moved among ratios and lines in a triangle as casually as we walk around the neighborhood. His was Perga, in Pamphylia (on the southwestern coast of Turkey), and later, Alexandria; and the time—about 240 B.C.—puts him in the generation after Archimedes.

  For the sake of ease we’ll give his proof algebraically, although this attenuates the magic of seeing it through the play of shapes.

  In ABC, with sides a, b, and c as shown, drop the median m meeting AB at D, and the altitude k from C to AB, meeting it at K. Let AK = t, so that KB = c − t.

  Talk about generalization’s nested cones—this theorem has a yet broader form, with D roaming as it will over AB. Matthew Stewart in Rothesay, off the distant Scottish coast, some time in the 1730s came up with a theorem for this more general setting that now proudly flies his name1—Stewart’s Theorem: If D is any point on c, dividing it into the segments m and n, and d is the line from C to D, then

  Stewart’s proof (you’ll find it in this chapter’s appendix) follows from applying to ADC and CDB the most famous of all extensions of the Pythagorean Theorem to any triangle: The Law of Cosines.

  You may think Apollonius’s Theorem frivolous and Stewart’s extension of it an effete variation. Yet neither is an idle étude—each is a significant sharpening of the mathematical outlook: a triangle’s three sides and the various lines that go
with them (altitudes, medians, and so on) depend on a few defining relations, one of which Apollonius gives for medians, and Stewart for the more general ‘Cevian’ (any line from the vertex to a triangle’s opposite side): a relation fundamental enough to determine all the rest for the triangle you see and the whole class of triangles congruent to it. How freedom lines up with necessity: the sort of acorn you carry off and hope to live on.

  This scary-sounding Law of Cosines, along with Driver’s Ed, is what definitively tells the recent middle school graduate that the world of higher learning has now begun. We know it in terms of trigonometry—the art of the ratio, developed by Arabic mathematicians:

  where γ is the angle between sides a and b.

  A millennium earlier, however, Euclid proved this with stunning simplicity. What’s more, his proof resurrects for us the Babylonian Box. You’ll remember we said in Chapter Two that the box, rightly thought of, showed that (a + b)2 = a2 + 2ab + b2.

  Showed—but didn’t prove, which Euclid, being a Greek, required. His proof (II.4 in his Elements) just adds a diagonal, making slews of equal

  angles where it crosses pairs of parallel lines, letting him deduce that what look like squares are, and that the remaining oblongs are indeed congruent rectangles.r

  With this result clipped to his belt and the Pythagorean Theorem in his pocket, Euclid now sets to work on the Law of Cosines. Here—in current terminology—is his elegant proof in the obtuse case (he follows it with a similar proof for the acute).

  One consequence of this result is the converse of the Pythagorean Theorem: if a2 + b2 = c2, the triangle whose sides these are is right. For if not, the angle in question being acute or obtuse would add a non-zero term to this sum (it needs of course no spirit from the trigonometric deep to tell us this: Euclid easily proves (I.48) the converse of the Pythagorean Theorem from the Theorem itself). Nothing happens in isolation: we will soon need this converse.

  You’ve just seen from the law of cosines not only that, but why, c2 is greater than a2 + b2 in an obtuse, less than it in an acute, and equal to it in a right triangle. With the change in viewpoint from the static to the dynamic, it’s tempting to hook this trio up to the paired inequalities of the angles: respectively.

  Edsger Dijkstra, among the last century’s leading pioneers in computer science, found a more succinct way of putting this (using ‘’ to mean ‘if and only if’):

  Looking for greater concision, he rewrote:

  For now he could rethink these expressions in terms of their signs—negative, zero, or positive—and (in the grip of this frenzy for condensing) abbreviate even ‘sign’ to the sign ‘sgn’, and so conclude:

  There’s our whole book so far, freeze-dried to half a line of squiggles.

  Dijkstra wasn’t content with a mere summary. In fact he seems not to have been content with most things, making him an enfant terrible to many, an awful child to more. One of his colleagues spoke of him as a giant, adding: “But intellectual giants are odd. They distort the usual laws of perspective: as you get closer to them, they seem smaller.” His oddness included indulging in a running fantasy about a company called Mathematics Inc., of which he was the chairman. It manufactured theorems, keeping their proofs a trade secret (it had also produced The Standard Proof of the Pythagorean Theorem, which outsold all the incompatible existing proofs).

  In a memorial for him, another colleague wrote: “Opinions, different from his, met with his greatest disapproval, and he related to them in a famously obnoxious manner.” Arrogance in computer science is apparently now measured in nano-Dijkstras. Yet while he cultivated rudeness toward the world, his small band of followers thought him compassionate, a man who would drop in unannounced of an evening to chat for an hour or two—a grammarian in sandals.

  In any case, Dijkstra now set out in reality to prove his Standardized Statement of the Pythagorean Theorem. He drew a picture of an obtuse triangle ABC with its obtuse angle, γ, at C:

  and since is negative.

  He then created two triangles, 1 and 3, similar to ABC and inside it, as shown, with an isosceles triangle 2 between them:

  He no sooner did this than he regretted it: “No cheers at all for that stage of the argument which forced us to resort to a picture. Pictures are almost unavoidably overspecific.” But he bravely went on: since 1 and 3 don’t cover ABC,

  Well, what do you make of all this? Is it just another, rather awkward, proof of the law of cosines, or something much more? Dijkstra wrote: “We have proved a theorem, say, 4 times as rich as [the usual formulation of the Pythagorean Theorem].” What exactly is this richness? Does it lie in having reduced the statement to a bit of code? Elsewhere Dijkstra said that “the traditional mathematician recognizes and appreciates mathematical elegance when he sees it. I propose to go one step further, and to consider elegance an essential ingredient of mathematics: if it’s clumsy, it’s not mathematics.”

  Is this mathematics? We come back to our discussion of elegance in Chapter Five, where (to put it in our present terms) we suggested that elegance and economy may have the same sign, but aren’t necessarily equal. Even if they were, all those omitted cases clutter up the background of this proof, and a mere word count, or inventory of its steps, shows it less economical than Euclid’s. But something much more important, we think, is involved. For all the statement’s concision, it has been torn from the history, associations, intuitions, and implications of the Pythagorean Theorem, and made into a cell in cyberspace that discloses nothing. It is as if you wanted to hear a quartet movement of Mozart’s and were played instead the chord from which its harmonies and melodies unfolded. Neither math nor music is an excursion in syntax, but in semantics.

  We’ll leave the last words to Dijkstra himself. He ends his proof with this sardonic, if not quite English, reflection: “I am in a paradoxical situation. I am convinced that of the people knowing the theorem of Pythagoras, almost no one can read the above without being surprised at least once. Furthermore, I think that all those surprises relevant (because telling about their education in reasoning). Yet I don’t know of a single respectable journal in which I could flog this dead horse.” Well, there was always Mathematics Inc.2

  A PLANE OTHER THAN THE REAL

  So precariously is mathematics balanced on the edge of tautology that the gentlest push can tumble it into the depths. Renaming is such a push (since it really stands for looking at one thing from two points of view), as when we call our triangle’s sides no longer a, b, and c but sin θ, cos θ, and 1,

  In the best tales, the door to the secret garden blends into the bricks of its wall, and tautologies are the bricks of mathematics. Look at the garden the miraculous Swiss mathematician Leonhard Euler fell into when he opened this door.3 He first factored the new equation into

  —a jaw-dropping move—from which he swiftly developed the infinite series you saw in Chapter Five:

  But there were orchids yet more fantastic growing here, which he and others cultivated, such as his astonishing formula,

  The garden we have found ourselves in is the complex plane, where so much that was unsettled on the real is completed, and symmetry restored. An nth degree polynomial, for example, has at most n real roots—but all n of them spread through the complex plane—and in patterns that would delight a Capability Brown: 1 has two real square roots, 1 and −1; three complex cube roots ; four complex fourth roots; and so on: exactly n complex nth roots, arranged as the vertices of a regular n-gon:

  These all are the offspring of . We will wander in this garden later, to collect some of its most exotic species: flowers whose corollas tunnel to another world.

  DIMENSIONS OTHER THAN TWO

  Because the Pythagorean Theorem concerns areas, we think of one plane or another as its natural home. But inquiry is restless and imagination boundless, so very soon you ask: what about the ‘body diagonal’ of a rectangular box?

  Draw in the hypotenuse e on the floor of this box. Then on the plane


  Was ever so wide an insight leapt with so little effort? Perhaps the unused energy you stored up for it has set off a wild surmise in your parietal cortex (where we do our counting): a sum of two squares in two dimensions, of three in three . . . But let your eye restrain your brain for a moment by looking back to seventeenth-century Ulm, where the Master Reckoner Johannes Faulhaber not only discovered a different spatial generalization of the Pythagorean Theorem, but saw how it would lead him to a deeper understanding of 666, which he already knew was divine rather than diabolic. While the margins of our minds are too narrow to contain Faulhaber’s kabbalistic proof, we will unveil to you his rectilinear pyramid.

  “Analogy”, said his contemporary Johannes Kepler, “is my most faithful teacher.” Moving by analogy from squares on the lengths of a right triangle’s sides to squares on the areas of an orthonormal pyramid’s faces, he came up with this beautiful insight: given such a pyramid (three of its edges meeting at right angles, as in the corner of a box), the sum of the squared areas of these faces (call them A, B, and C)